Game theory has shown that prisoner's dilemma requires people to rationally renege; unless it is iterated, in which case it is easy to be good (use tit-for-tat). This explains why long term cafe customers are valued but used car salesmen and real-estate are considered the scum of the earth. Without iteration it is better to be a bastard than honest.
Now consider the case of private health insurance. You pick a company based on a belief in future service (don't even try to get health insurance when you're sick). You get one chance to pick the right company. Furthermore, there is an information asymmetry: health insurance companies have detailed analysis of customers likely woes, but customers know little more than the companies' logo colour scheme. So it's a non-iterated PD problem.
So it is in the interest of a profit making health care company to give the appearance of caring, whilst screwing as many people as possible. They get little or no incentive to do otherwise (in the worst case the company plays a shell game and moves to a new logo when their reputation begins to suffer). People rarely even discuss the quality of their health care (americans in particular seem very averse to discussing non-life threatening health problems), let alone perform cost benefit analysis.
Furthermore, health is something that each individual considers of highest value. Thus they are willing to pay vastly more than is rational for the service they get (and of course the most successful health companies have discovered this).
Why doesn't public health care suffer the same problem? Because its incentives are different. It seeks to maximise government money and staffing. And the government, who can draw in the same (or more) resources for analysis, provides the countering balancing force of cost cutting and cost effectiveness analysis. To see this in action, look at the recent kerfuffle over unneeded CAT scans in the US system.
Sure enough, the cost per person per year of health systems around the world match this reality. The US pays $6500/py for a less effective system than the UK who pay $2500/py. My personal experience of the US and Australia has reflected this - in Australia I got elective surgery for free (paid for by the public system) immediately, in the US I can't even get a GP visit without lengthy waiting and poor service. Yet HP paid $16k/pa for my health care in the US, compared to approximately $3k/pa taken out of my income tax in Australia.
One might counter that we can solve these private health problems with ombudsmen and independent review. But those apply equally to public health, without the profit syphoning.